NAG CL Interface
f12agc (real_band_solve)
Note: this function uses optional parameters to define choices in the problem specification. If you wish to use default
settings for all of the optional parameters, then the option setting function f12adc need not be called.
If, however, you wish to reset some or all of the settings please refer to Section 11 in f12adc for a detailed description of the specification of the optional parameters.
1
Purpose
f12agc is the main solver function in a suite of functions consisting of
f12adc,
f12afc and
f12agc. It must be called following an initial call to
f12afc and following any calls to
f12adc.
f12agc returns approximations to selected eigenvalues, and (optionally) the corresponding eigenvectors, of a standard or generalized eigenvalue problem defined by real banded nonsymmetric matrices. The banded matrix must be stored using the LAPACK
column ordered
storage format for real banded nonsymmetric
(see
Section 3.4.4 in the
F07 Chapter Introduction).
2
Specification
void 
f12agc (Integer kl,
Integer ku,
const double ab[],
const double mb[],
double sigmar,
double sigmai,
Integer *nconv,
double dr[],
double di[],
double z[],
double resid[],
double v[],
double comm[],
Integer icomm[],
NagError *fail) 

The function may be called by the names: f12agc, nag_sparseig_real_band_solve or nag_real_banded_sparse_eigensystem_sol.
3
Description
The suite of functions is designed to calculate some of the eigenvalues, $\lambda $, (and optionally the corresponding eigenvectors, $x$) of a standard eigenvalue problem $Ax=\lambda x$, or of a generalized eigenvalue problem $Ax=\lambda Bx$ of order $n$, where $n$ is large and the coefficient matrices $A$ and $B$ are banded, real and nonsymmetric.
Following a call to the initialization function
f12afc,
f12agc returns the converged approximations to eigenvalues and (optionally) the corresponding approximate eigenvectors and/or an orthonormal basis for the associated approximate invariant subspace. The eigenvalues (and eigenvectors) are selected from those of a standard or generalized eigenvalue problem defined by real banded nonsymmetric matrices. There is negligible additional computational cost to obtain eigenvectors; an orthonormal basis is always computed, but there is an additional storage cost if both are requested.
The banded matrices
$A$ and
$B$ must be stored using the LAPACK column ordered storage format for banded nonsymmetric matrices; please refer to
Section 3.4.2 in the
F07 Chapter Introduction for details on this storage format.
f12agc is based on the banded driver functions
dnbdr1 to
dnbdr6 from the ARPACK package, which uses the Implicitly Restarted Arnoldi iteration method. The method is described in
Lehoucq and Sorensen (1996) and
Lehoucq (2001) while its use within the ARPACK software is described in great detail in
Lehoucq et al. (1998). An evaluation of software for computing eigenvalues of sparse nonsymmetric matrices is provided in
Lehoucq and Scott (1996). This suite of functions offers the same functionality as the ARPACK banded driver software for real nonsymmetric problems, but the interface design is quite different in order to make the option setting clearer and to combine the different drivers into a general purpose function.
f12agc, is a general purpose function that must be called following initialization by
f12afc.
f12agc uses options, set either by default or explicitly by calling
f12adc, to return the converged approximations to selected eigenvalues and (optionally):

–the corresponding approximate eigenvectors;

–an orthonormal basis for the associated approximate invariant subspace;

–both.
Please note that for ${\mathbf{Generalized}}$ problems, the ${\mathbf{Shifted\; Inverse\; Imaginary}}$ and ${\mathbf{Shifted\; Inverse\; Real}}$ inverse modes are only appropriate if either $A$ or $B$ is symmetric semidefinite. Otherwise, if $A$ or $B$ is nonsingular, the ${\mathbf{Standard}}$ problem can be solved using the matrix ${B}^{1}A$ (say).
4
References
Lehoucq R B (2001) Implicitly restarted Arnoldi methods and subspace iteration SIAM Journal on Matrix Analysis and Applications 23 551–562
Lehoucq R B and Scott J A (1996) An evaluation of software for computing eigenvalues of sparse nonsymmetric matrices Preprint MCSP5471195 Argonne National Laboratory
Lehoucq R B and Sorensen D C (1996) Deflation techniques for an implicitly restarted Arnoldi iteration SIAM Journal on Matrix Analysis and Applications 17 789–821
Lehoucq R B, Sorensen D C and Yang C (1998) ARPACK Users' Guide: Solution of Largescale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods SIAM, Philidelphia
5
Arguments

1:
$\mathbf{kl}$ – Integer
Input

On entry: the number of subdiagonals of the matrices $A$ and $B$.
Constraint:
${\mathbf{kl}}\ge 0$.

2:
$\mathbf{ku}$ – Integer
Input

On entry: the number of superdiagonals of the matrices $A$ and $B$.
Constraint:
${\mathbf{ku}}\ge 0$.

3:
$\mathbf{ab}\left[\mathit{dim}\right]$ – const double
Input

Note: the dimension,
dim, of the array
ab
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\times \left(2\times {\mathbf{kl}}+{\mathbf{ku}}+1\right)\right)$ (see
f12afc).
On entry: must contain the matrix
$A$ in LAPACK columnordered banded storage format for nonsymmetric matrices (see
Section 3.4.4 in the
F07 Chapter Introduction).

4:
$\mathbf{mb}\left[\mathit{dim}\right]$ – const double
Input

Note: the dimension,
dim, of the array
mb
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\times \left(2\times {\mathbf{kl}}+{\mathbf{ku}}+1\right)\right)$ (see
f12afc).
On entry: must contain the matrix
$B$ in LAPACK columnordered banded storage format for nonsymmetric matrices (see
Section 3.4.4 in the
F07 Chapter Introduction).

5:
$\mathbf{sigmar}$ – double
Input

On entry: if one of the
${\mathbf{Shifted\; Inverse\; Real}}$ modes (see
f12adc) have been selected then
sigmar must contain the real part of the shift used; otherwise
sigmar is not referenced.
Section 4.3.4 in the
F12 Chapter Introduction describes the use of shift and inverse transformations.

6:
$\mathbf{sigmai}$ – double
Input

On entry: if one of the
${\mathbf{Shifted\; Inverse\; Real}}$ modes (see
f12adc) have been selected then
sigmai must contain the imaginary part of the shift used; otherwise
sigmai is not referenced.
Section 4.3.4 in the
F12 Chapter Introduction describes the use of shift and inverse transformations.

7:
$\mathbf{nconv}$ – Integer *
Output

On exit: the number of converged eigenvalues.

8:
$\mathbf{dr}\left[\mathit{dim}\right]$ – double
Output

Note: the dimension,
dim, of the array
dr
must be at least
${\mathbf{nev}}+1$ (see
f12afc).
On exit: the first
nconv locations of the array
dr contain the real parts of the converged approximate eigenvalues. The number of eigenvalues returned may be one more than the number requested by
nev since complex values occur as conjugate pairs and the second in the pair can be returned in position
${\mathbf{nev}}+1$ of the array.

9:
$\mathbf{di}\left[\mathit{dim}\right]$ – double
Output

Note: the dimension,
dim, of the array
di
must be at least
${\mathbf{nev}}+1$ (see
f12afc).
On exit: the first
nconv locations of the array
di contain the imaginary parts of the converged approximate eigenvalues. The number of eigenvalues returned may be one more than the number requested by
nev since complex values occur as conjugate pairs and the second in the pair can be returned in position
${\mathbf{nev}}+1$ of the array.

10:
$\mathbf{z}\left[{\mathbf{n}}\times {\mathbf{nev}}\right]$ – double
Output

On exit: if the default option
${\mathbf{Vectors}}=\text{Ritz}$ has been selected then
z contains the final set of eigenvectors corresponding to the eigenvalues held in
dr and
di. The complex eigenvector associated with the eigenvalue with positive imaginary part is stored in two consecutive array segments. The first segment holds the real part of the eigenvector and the second holds the imaginary part. The eigenvector associated with the eigenvalue with negative imaginary part is simply the complex conjugate of the eigenvector associated with the positive imaginary part.
For example, if ${\mathbf{di}}\left[0\right]$ is nonzero, the first eigenvector has real parts stored in locations
${\mathbf{z}}\left[\mathit{i}\right]$, for $\mathit{i}=0,1,\dots ,{\mathbf{n}}1$ and imaginary parts stored in
${\mathbf{z}}\left[\mathit{i}\right]$, for $\mathit{i}={\mathbf{n}},\dots ,2{\mathbf{n}}1$.

11:
$\mathbf{resid}\left[\mathit{dim}\right]$ – double
Input/Output

Note: the dimension,
dim, of the array
resid
must be at least
${\mathbf{n}}$ (see
f12afc).
On entry: need not be set unless the option
${\mathbf{Initial\; Residual}}$ has been set in a prior call to
f12adc in which case
resid must contain an initial residual vector.
On exit: contains the final residual vector.

12:
$\mathbf{v}\left[{\mathbf{n}}\times {\mathbf{ncv}}\right]$ – double
Output

On exit: if the option
${\mathbf{Vectors}}$ (see
f12adc) has been set to Schur or Ritz then the first
nconv sections of
v, of length
$n$, will contain approximate Schur vectors that span the desired invariant subspace.
The $i$th Schur vector is stored in locations
${\mathbf{v}}\left[{\mathbf{n}}\times \left(\mathit{i}1\right)+\mathit{j}1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{nconv}}$ and $\mathit{j}=1,2,\dots ,n$.

13:
$\mathbf{comm}\left[\mathit{dim}\right]$ – double
Communication Array

Note: the actual argument supplied
must be the array
comm supplied to the initialization routine
f12afc.
On entry: must remain unchanged from the prior call to
f12adc and
f12afc.
On exit: contains no useful information.

14:
$\mathbf{icomm}\left[\mathit{dim}\right]$ – Integer
Communication Array

Note: the actual argument supplied
must be the array
icomm supplied to the initialization routine
f12afc.
On entry: must remain unchanged from the prior call to
f12adc and
f12afc.
On exit: contains no useful information.

15:
$\mathbf{fail}$ – NagError *
Input/Output

The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
 NE_BAD_PARAM

On entry, argument $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_COMP_BAND_FAC

Failure during internal factorization of complex banded matrix. Please contact
NAG.
 NE_COMP_BAND_SOL

Failure during internal solution of complex banded matrix. Please contact
NAG.
 NE_INITIALIZATION

Either the initialization function has not been called prior to the first call of this function or a communication array has become corrupted.
 NE_INT

On entry, ${\mathbf{kl}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{kl}}\ge 0$.
On entry, ${\mathbf{ku}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{ku}}\ge 0$.
The maximum number of iterations $\text{}\le 0$, the option ${\mathbf{Iteration\; Limit}}$ has been set to $\u2329\mathit{\text{value}}\u232a$.
 NE_INTERNAL_EIGVAL_FAIL

Error in internal call to compute eigenvalues and corresponding error bounds of the current upper Hessenberg matrix. Please contact
NAG.
 NE_INTERNAL_EIGVEC_FAIL

Error in internal call to compute eigenvectors. Please contact
NAG.
 NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
 NE_INVALID_OPTION

On entry, ${\mathbf{Vectors}}=\text{Select}$, but this is not yet implemented.
 NE_MAX_ITER

The maximum number of iterations has been reached. The maximum number of $\text{iterations}=\u2329\mathit{\text{value}}\u232a$. The number of converged eigenvalues $\text{}=\u2329\mathit{\text{value}}\u232a$.
 NE_NO_ARNOLDI_FAC

Could not build an Arnoldi factorization. The size of the current Arnoldi factorization $=\u2329\mathit{\text{value}}\u232a$.
 NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
 NE_NO_SHIFTS_APPLIED

No shifts could be applied during a cycle of the implicitly restarted Lanczos iteration.
 NE_OPT_INCOMPAT

The options ${\mathbf{Generalized}}$ and ${\mathbf{Regular}}$ are incompatible.
 NE_REAL_BAND_FAC

Failure during internal factorization of real banded matrix. Please contact
NAG.
 NE_REAL_BAND_SOL

Failure during internal solution of real banded matrix. Please contact
NAG.
 NE_SCHUR_EIG_FAIL

During calculation of a real Schur form, there was a failure to compute a number of eigenvalues. Please contact
NAG.
 NE_SCHUR_REORDER

The computed Schur form could not be reordered by an internal call. Please contact
NAG.
 NE_TRANSFORM_OVFL

Overflow occurred during transformation of Ritz values to those of the original problem.
 NE_ZERO_EIGS_FOUND

The number of eigenvalues found to sufficient accuracy is zero.
 NE_ZERO_INIT_RESID

The option
${\mathbf{Initial\; Residual}}$ was selected but the starting vector held in
resid is zero.
 NE_ZERO_SHIFT

The option
${\mathbf{Shifted\; Inverse\; Imaginary}}$ has been selected and
${\mathbf{sigmai}}=\text{}$ zero on entry;
sigmai must be nonzero for this mode of operation.
7
Accuracy
The relative accuracy of a Ritz value,
$\lambda $, is considered acceptable if its Ritz estimate
$\le {\mathbf{Tolerance}}\times \left\lambda \right$. The default
${\mathbf{Tolerance}}$ used is the
machine precision given by
X02AJC.
8
Parallelism and Performance
f12agc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f12agc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
None.
10
Example
This example constructs the matrices $A$ and $B$ using LAPACK band storage format and solves $Ax=\lambda Bx$ in shifted imaginary mode using the complex shift $\sigma $.
10.1
Program Text
10.2
Program Data
10.3
Program Results